Large solutions of semilinear elliptic equations under the Keller–Osserman condition

We consider the equation Δu=p(x)f(u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0,∞), satisfies f(0)=0, f(s)>0 for s>0 and the Keller–Osserman condition where . We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation.